Definition 2.5 (\({\rm BV}\)  space) 
A function \(f\in L^1(X,\mathfrak {m})\)  belongs to the space \({\rm BV}(X,\mathsf {d},\mathfrak {m})\)  of functions of bounded variation (see [1]}, [2]}) if there exists a sequence \((f_n)_{n\in \mathbb {N}}\in \mathsf {Lip}_{loc}(X)\)  converging to \(f\)  in \(L^1(X,\mathfrak {m})\)  and such that
\(\limsup _{n\rightarrow \infty }\int _X \mathrm {lip}(f_n)\, \mathsf {d}\mathfrak {m}<+\infty \, .\) 
